Deformations of Batalin-Vilkovisky algebras

TL;DR

This paper generalizes Batalin-Vilkovisky algebras by exploring graded commutative algebras with higher-order differential operators, leading to new homotopy structures and conjectures extending Kontsevich's formality theorem.

Contribution

It introduces a framework for Batalin-Vilkovisky algebras up to homotopy using higher-order differential operators and proposes a conjecture extending the formality theorem.

Findings

Higher-order differential operators induce L$_mbda$-algebra structures.

A new definition of Batalin-Vilkovisky algebra up to homotopy is provided.

Conjecture extends Kontsevich's formality theorem to BV algebras.

Abstract

We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A, an operator of an order higher than 2 (Koszul-Akman definition) leads to the structure of a strongly homotopy Lie algebra (L$_\infty$-algebra) on A. This allows us to give a definition of a Batalin-Vilkovisky algebra up to homotopy. We also make a conjecture which is a generalization of the formality theorem of Kontsevich to the Batalin-Vilkovisky algebra level.